On the cardinality of the message space in sender-receiver games

We study sender-receiver games in which a privately informed sender sends a message to N receivers, who then take an action. The sender's type space T has finite cardinality (i.e., vertical bar T vertical bar < infinity). We show that every equilibrium payoff vector (resp. every Pareto efficient equilibrium payoff vector) is achieved by an equilibrium in which the sender sends at most vertical bar T vertical bar + N (resp. vertical bar T vertical bar + N - 1) messages with positive probability. We also show that such bounds do not exist when two privately informed senders simultaneously send a message to a receiver. (c) 2020 Elsevier B.V. All rights reserved.